Water vapor is the basic parameter used to describe atmospheric conditions. It is rarely contained in the atmosphere during the water cycle, but it is the most active element in rapid space–time changes. Measuring and monitoring the distribution and quantity of water vapor is a necessary task. GPS tomography is a powerful means of providing high spatiotemporal resolution of water vapor density. In this paper, a spatial structure model of a humidity field is constructed using voxel nodes, and new parameterizations for acquiring data about water vapor in the troposphere via GPS are proposed based on inverse distance weighted (IDW) interpolation. Unlike the density of water vapor that is constant within a voxel, the density at a certain point is determined by IDW interpolation. This algorithm avoids the use of horizontal constraints to smooth voxels that are not crossed by satellite rays. A prime number decomposition (PND) access order scheme is introduced to minimize correlation between slant wet delay (SWD) observations. Four experimental schemes for GPS tomography are carried out in dry weather from 2 to 8 August 2015 and rainy days from 9 to 15 August 2015. Using 14 days of data from the Hong Kong Satellite Positioning Reference Station Network (SatRef), the results indicate that water vapor density derived from 4-node methods is more robust than that derived from that of 8 nodes or 12 nodes, or that derived from constant refractivity schemes and the new method has better performance under stable weather conditions than unstable weather (e.g., rainy days). The results also indicate that an excessive number of interpolations in each layer reduce accuracy. However, the accuracy of the tomography results is gradually reduced with increases in altitude below 7000 m. Moreover, in the case of altitudes between 7000 m and the upper boundary layer, the accuracy can be improved by a boundary constraint.

Based on the GPS meteorology technique (Bevis et al., 1992), two experiments were carried out to assist the development of GPS integrated water vapor (IWV) inversion: GPS/STORM (Rocken et al., 1995) and the GPS–Winter Icing and Storms Project experiment (Gutman et al., 1994). Both experiments showed that GPS is a cost-effective and reliable means of continuously monitoring IWV, with accuracies comparable to water vapor radiometers (WVRs) and radiosondes (RSs). With the development of the GPS–IWV network, many studies have shown that IWV determined by GPS can achieve an accuracy of 1–2 mm (Duan et al., 1996; Emardson et al., 2000; Liang et al., 2015; Niell, 2001; Raja et al., 2008; Rocken et al., 1993). GPS/IWV has been applied to improve the quality of numerical weather prediction (NWP) models and most recently GNSS has been used for nowcasting and help with the predictability of severe weather (De Haan et al., 2009; Gendt et al., 2004; Ichikawa et al., 2012; Rohm et al., 2014; Smith et al., 2007). It is a powerful source of continuous (10–30 min temporal resolution) integrated water vapor determination for climate studies, weather prediction, and hazard mitigation. However, IWV is a measure of the total amount of water vapor above a certain station, and it cannot provide information on the vertical distribution of water vapor.

In order to meet the demand, GNSS (Global Navigation Satellite System) water vapor tomography has appeared as a promising method of providing information on the four-dimensional distribution of the water vapor in the troposphere. By using extensive slant wet delay (SWD) data collected from a network of GNSS receivers, a four-dimensional humidity field with high spatial and temporal resolution can be reconstructed from tomography. Over the past decade, numerous studies demonstrated the potential of GNSS tomography to retrieve 4-D humidity fields. Unfortunately, because the quality of the reconstructed profiles is limited by the constellation of GNSS satellites, the geographic distribution of ground-based receivers, and observation errors (Chen and Liu, 2014; Rohm et al., 2014; Shangguan et al., 2013; Yao et al., 2016), some voxels may not be crossed by any signal during a tomographic process. Consequently, this will lead to an ill-posed inverse problem with incomplete input data.

The methods for solving the abovementioned problems can be broadly divided into four categories: (1) enhancement of the precision of SWD using the “zero differences” (ZDs) technique (Alber et al., 2000; Iwabuchi, 2004; Seko et al., 2004); (2) addition of constraint conditions to tomographic models, e.g., horizontal, vertical, and boundary constraint conditions (Flores et al., 2000; Hirahara, 2000; Perler, 2011; Rohm and Bosy, 2009; Seko et al., 2000; Song et al., 2006); (3) usage of additional extra observations through RINEX met files, zenith wet delay, WVR, RS, and voxel-optimized regional water vapor tomography (Bi et al., 2006; Chen and Liu, 2014; Jiang et al., 2014; Rocken et al., 1993; Rohm et al., 2014; Yao et al., 2016); and (4) new algorithms to improve inversion quality, such as singular value decomposition (SVD), the wet refractivity Kalman filter (KF), algebraic reconstruction techniques (ARTs), and the parameterization of voxels (volumetric pixels) based on trilinear and spline functions (Bender et al., 2011; Flores et al., 2001; Gradinarsky, 2002; Gradinarsky and Jarlemark, 2004; Nilsson and Gradinarsky, 2006; Rohm et al., 2013; Shangguan et al., 2013). At present, we are focused on replacing divided voxel-based traditional methods with new, parameterized approaches.

In this paper, we introduce a parameterized approach based on horizontal IDW interpolation, in which only the vertical constraint is used to ensure that the model is compliant with the features of water distribution in the troposphere. The new algorithm is analyzed using data from SatRef. The experiment mainly analyzes and discusses the influence that different numbers of voxel nodes have on the results of GPS water vapor tomography. Moreover, we address situations in which some voxels are not crossed by any signals. However, grid points not used in any interpolation should not generally be avoided. In fact, this case often occurs in lower-level layers with the four-node method, and occurs less often when many points are included in the interpolation. We also address “empty” grids using the inverse distance weighted (IDW) interpolation. The values of an “empty” grid are estimated by calculating the “non-null” grids around it.

Atmospheric refraction delay, including ionospheric delay and neutral
atmospheric delay, is a major source of GPS positioning errors. Ionospheric
delay (dispersive delay) can be corrected by using dual-frequency linear
combinations. Neutral atmospheric delay, independent of frequency, is an area
of great research interest in geosciences and meteorology. The slant total delay (STD)
can be expressed by (Bevis et al., 1992)

The delay caused by atmospheric refractivity contains the slowly varying
delays caused by dry air and the rapidly varying delays caused by water
vapor. This characteristic can be represented by the following formula
(Thayer and Gordon, 1974):

IDW parameterization assumes that the troposphere is divided into a number of vertical layers (see Fig. 1). SWD produced by one signal (blue line) is discretely modeled as the sum of the values at the points of intersection between straight lines and vertical layers, multiplied by the corresponding distance. There are no vertical variations assumed within a layer. In our study, the true path of the signal between the satellite and the receiver will be replaced by a straight line, and the vertical structure of nonuniform layers is used.

The SWD of a signal propagation path can be expressed by the formula:

Vertical structure of IDW tomography model and the signal rays (blue line) crossing the vertical layers. The value of SWV divides into the value of water vapor at pierce points (red points).

The IDW parametrical method explicitly implements the assumption that the
values of wet refractivity in close proximity are more alike compared to
those that are farther away. It is clear that the wet refractivity at an
intersection can be calculated based on a weighted average of the values at
voxel nodes that surround it. The most common method to define intersection
value is to utilize a limited number of voxel nodes on the same layer to
express it. For example, in Fig. 2, four adjacent nodes (

A case showing an IDW tomography model in a certain layer. Black
points are voxel nodes used for predicting the value of the pierce point (red
point); the gray wireframe indicates the layer used in the model, and the
length of the double arrows represents the distance between the point

Based on IDW interpolation,

In order to reconstruct the characteristics of water vapor distribution in a
vertical direction, the vertical constraint will be utilized. We will assume
an exponential law (Davis et al., 1993), which can be used to represent an
average water vapor profile:

By combining all the SWD observations (Eq. 7) and vertical constraints
(Eq. 10), the solution model of IDW tomography in a matrix form is given as

Algebraic reconstruction techniques (ARTs) have been successfully used to reconstruct the humidity field (Bender et al., 2011; Chen et al., 2014). An advantage of ARTs is that they have high numerical stability even under adverse conditions. Moreover, it is relatively easy to incorporate prior knowledge into the reconstruction process.

The solution model of IDW tomography is defined by

There are other ARTs, such as the symmetric method (Björck and Elfving, 1979) and the randomized method (Strohmer and Vershynin, 2007), that can distinguish according to the order in which the rows are processed. Therefore, the access order of the ART has a significant effect on its practical performance (Herman, 1980). The importance of access order has been recognized in medical applications of ART (Mueller et al., 1997; Herman and Meyer, 1993; Hounsfield, 1976; Robb et al., 1974).

In our study, an access order scheme based on prime number decomposition (PND) will be proposed. It is desirable to order the SWD observations such that subsequently applied SWDs are largely uncorrelated (Ding et al., 2016). This means that consecutively available SWDs must have significantly different values, because the value of SWD is determined by the azimuth and elevation angles of a signal. If the SWDs in a set have similar values solved by ART, the results tend to move away from the desired solution, which delays convergence. To summarize, the principle of SWD access order is that in a subsequence of iterations of Eq. (15), steps should be as independent as possible from the previous steps.

Following the decorrelation principle, the PND access order scheme (Herman
and Meyer, 1993) is presented in this section. In the first step, we sort all
elements of the SWD vector (in Eq. 10) in descending order; the sorted
equations will be numerated from 0 to

Prime number decomposition access order.

SatRef is a local satellite positioning system covering the extent of Hong
Kong. The network consists of 18 continuously operating reference stations
(CORSs); 14 of them were used in this study. The horizontal and vertical
station distributions are presented in Fig. 3a and b, respectively. The area
of investigation ranges from 113.749 to 114.474

In the methodology described in this section, samples of GPS and surface
meteorological data are obtained in 30 s intervals. The data from the Hong
Kong network (Fig. 3a) were processed with GAMIT software version 10.5 on the
basis of double differences. The tropospheric zenith delays and the gradient
parameters were estimated by a piecewise linear (PWL) model with resolution
of 10 min. SWDs are calculated with temporal resolution of 10 and 30 min
data of SWDs are applied to building humidity field. A cut-off elevation
angle of 10

Comparison of HKKP's radiosonde data (blue dot) with results of water vapor tomography schemes (red lines) at 00:00 and 12:00 UTC, respectively (DOY 221 2015).

Based on the number of voxel nodes

4nodes: four voxel nodes (neighbors) from the voxel node model are used when calculating a wet refractivity value for the piercing point.

8nodes: eight voxel nodes are used when calculating a wet refractivity value for the piercing point.

12nodes: 12 voxel nodes are used when calculating a wet refractivity value for the piercing point.

Constant: the refractivity is constant within a voxel when calculating a wet refractivity for the piercing point.

The four solutions are compared with radiosonde (RS) data recorded in Hong Kong. Because of limited space, only a representative example from 9 August 2015 (DOY 221) 00:00 and 12:00 UTC is shown (see Fig. 4). It is clear that all of the water vapor profiles (red lines) reconstructed by the tomographic solutions accord with the radiosonde data. At 00:00 UTC, all profiles and the scatter of radiosonde data decrease exponentially with increases in height. However, at 12:00 UTC, radiosonde data indicate disturbances in water vapor density and are thus less stable than at 00:00 UTC. In each layer (Fig. 3b), the mean water vapor density is computed. It indicates that it is difficult to retrieve humidity data with high vertical resolution (e.g., radiosonde) because of the limited accuracy of the tomography solution. However, consistency is maintained with the changing trends of the radiosonde data. It is difficult to determine which solution is best suited for GPS water vapor tomography from the statistics of these data (Table 2). These solutions have their respective advantages. The 4nodes parameterization presents more accurate results than others in terms of RMSE. The interquartile range (IQR), to some extent, reflects the discreteness of datasets. Based on the IQR values (Table 2) and the water vapor density scatter plots (Fig. 5a–d), it is clear that the tomographic solutions computed by 4nodes parameterizations are more concentrated than those computed by the others. However, compared with the other methods, the 12nodes tomography has the smallest bias (Table 2) and the highest discretization.

Statistics showing the differences between the four schemes and water vapor measurements constructed by HKKP's radiosonde data for 7 days (DOY 221–227) in 2015.

Tomographic solutions compared with radiosonde data.

Statistics of the humidity differences between tomographic solutions
and radiosonde data in each layer from DOY 221 to 227 2015.

The statistical characteristics of the differences between the four schemes
and the radiosonde data are also presented by box plots (Fig. 5e). They
contain five characteristic values: the first and third quartiles (Q1 and Q3)
are located at the bottom and top of the box; the second quartile (Q2) is
located inside the box (the median) and at the ends of the whiskers
(upper and lower bound) at Q1

IQR is the difference between the Q3 and Q1 quartiles. It is important because IQR represents the spread of data and, unlike the total range, it is not affected by outliers. Q2 is the measure of central tendency and is in good accordance with the bias (as the statistics in Table 2 indicate). Upper and lower bounds can be used to identify outliers. The constant scheme has the smallest number of outliers, but it also has maximum bounds. The 8nodes and 12nodes schemes have similar bounds. However, if we use the bounds of 4nodes, the numbers of outliers in the 8nodes, 12nodes, and constant schemes are 66, 30, and 30, respectively. Therefore, comparison of 4nodes parameterization with the other schemes shows that it has a relatively minimal number of outliers.

Characteristic values of box plots of the four schemes and water vapor measurements constructed by HKKP's radiosonde data for 7 days (DOY 221–227) in 2015.

The above analyses are based on total statistics of data from a 7-day period. However, the precision of water vapor tomography is highly influenced by the vertical structure of the humidity field. For example, the same tomographic error value in different layers has different qualities. Therefore, total statistics cannot provide a comprehensive analysis of the accuracy of tomographic reconstruction. The relative index is introduced for tomographic stratification analysis in Fig. 7. We adopt nonuniform layers within the interval from 500 to 3800 m. The box plots (Fig. 6a–d) reveal that the errors of 4nodes and 8nodes in each layer are, in general, more concentrated than those of 12nodes and the constant method. The results accord with the total statistical analysis, which was illustrated in a previous section.

Statistics of the humidity differences between tomographic
solutions and radiosonde data in each layer from DOY 221 to 227 2015.

A comparison of the three parameterizations in terms of RMSE (Fig. 6e–h) and
relative error (Fig. 6i–l) shows that 4nodes has relatively small values for these two parameters. However, by
observing the changes between RMSE and relative error, opposite tendencies
emerge in total parameterizations. For example, between 4000 and 7000 m, the
RMSE of the constant method decreased from 1.749 to 1.078 g m

Four methods are used again in this period. All of the water vapor profiles reconstructed by the tomographic solutions are in good agreement with the radiosonde data. To avoid repetition of conclusions, this section focuses on the difference between experiments during rainy days (DOY 221–227) and dry weather (DOY 214–220). The results of RMSE in experiment 2 (Table 4) increase accuracy by more than 2 times compared to that in experiment 1 (Table 2). In terms of bias and IQR, consistency is maintained with the case of RMSE. This means that the new method has better performance under stable weather conditions than unstable weather (e.g., rainy days). However, by comparing the results of outliers in both experiments, the main reasons for outliers in Table 4 having a remarkable increase are due to experiment 2 having smaller bound than those of experiment 1.

Statistics and characteristic values of box plots showing the differences between the four schemes and water vapor measurements constructed by HKKP's radiosonde data for 7 days (DOY 214–220) in 2015.

Comparing four methods in experiment 2, 4nodes has the smallest value of RMSE
and number of outliers. Similarly, 4nodes is more concentrated than other
methods. This case is reflected in the value of IQR (

RMSE and relative error in each layer (Fig. 7) have good agreement with experiment 1. It is show that as altitude increased, opposite tendencies emerged between the RMSE of each layer and the corresponding relative error. In the top layers, a boundary constraint condition dramatically reduces the RMSE and relative errors.

In this paper, a new GPS tomographic parameterization approach based on IDW interpolation is proposed. This approach can reconstruct a humidity field without using horizontal constraints and prevent situations in which some voxels are not crossed by satellite paths. On the other hand, instead of dividing the troposphere into several layers with identical heights, the vertical structure of the tomography model adopts nonuniform layers to satisfy inherent characteristics of water vapor distribution and to lower the effect of the difference magnitude between the calculated tomographic results. In order to minimize correlation in projection access, a PND access order scheme is developed to order the SWV observations such that subsequently applied SWV values are largely uncorrelated. Based on the number of voxel nodes selected for IDW interpolation, and the constant method that makes refractivity constant within a voxel, four schemes are designed to retrieve water vapor density into voxel nodes. In addition, a vertical constraint is adopted to ensure the characteristics of vertical water vapor distribution.

Tomographic experiments using GPS data collected over Hong Kong from
DOY 214 to 227 2015 validate our proposed GPS tomography-based approach. We
discuss and analyze 4nodes, 8nodes, and 12nodes methods as well as a constant
method. For the overall dataset, the 4nodes method offers the highest
accuracy compared to the other three methods, with perturbations of
1.627 g m

In future studies, the horizontal structure of the humidity field must be improved by adjusting the node position in each layer to fit the distribution of satellite rays. Flexible layout is the advantage of the voxel node model. In the future, the fusion of GNSS and external measurements from other sensors in the GPS tomography system will be a potential means to enhance the stability and reliability of water vapor tomography and to decrease tomography intervals.

GNSS data in RINEX format for GPS tomography from 2 to 15 August 2015 can be
freely downloaded from

The authors declare that they have no conflict of interest.

This study is supported by the National Natural Science Foundation of China (no. 41504032), the Natural Science Foundation of Jiangsu Province (BK20150175), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (no. 20130095110022). The authors acknowledge the Survey and Mapping Office (SMO) of the Lands Department for providing GNSS data in RINEX format from the Hong Kong Satellite Positioning Reference Station Network (SatRef). We also wish to show our appreciation to King's Park Meteorological Station for providing the radiosonde data. We thank the Department of Earth Atmospheric and Planetary Sciences, MIT, for providing GAMIT/GLOBK software. We are grateful to the editor and the reviewer team for their valuable comments, which helped us to improve the paper. The topical editor, M. Salzmann, thanks K. Zhang and T. Nilsson for help in evaluating this paper.